hey yo what's up you guys thank you for joining me again now in this video we're going to be covering chapter 8 which is Factor now in this part one video of this chapter we are going to be covering the basics of vector as well as addition and subtraction of factor so without further Ado let us get started [Music] foreign so before we start with the questions let us go through the basics of vector some of you may not be familiar with Vector but no worries Vector I'm sure some of you all would have learned this in physics here learn about the difference between vector quantity and scalar quantity okay scalar quantity so vector quantity means it has magnitude n Direction okay I'll explain more further Okay so Scala is it only has magnitude doesn't have Direction so for example height okay if you talk about someone's height there is no Direction involved right it's just how tall are you correct so that falls under scalar quantity or for example area there is no Direction in wolf in area correct so that falls under a scalar quantity but if you talk about Force you want to push something so when you apply Force Base Direction correct if you apply One Direction it goes to other direction correct so force in force Direction matters so Force falls under vector or you can say velocity or displacement so some of you all might ask I mean if you learn in physics you would have known that what's the difference between speed and velocity because why speed falls under scalar but velocity falls under Vector both seem the same correct both tells you tells you the speed but in velocity Direction methods why so for example let's say if I say I'm cycling okay you let's say you are cycling you are cycling from point A to point B okay you're cycling from point A to point B so if you're moving from point A to point B at say let's say your speed is uh 10 kilometer or maybe 10 meter per second okay let's say you're traveling at the speed of 10 meter per second so in this case the speed and the velocity is going to be the same which is 10 meter per second but if you're moving back okay now you're moving opposite direction when you're moving opposite direction your speed is still going to be the same 10 meter per second because it doesn't matter Direction doesn't matter so the speed is going to be the same but your velocity is going to be negative 10 meter per second negative 10 why because in velocity direction methods okay same goes to the difference between distance and what's a displacement okay I'm teaching physics now so distance does not take uh Direction into account so it falls under scalar quantity and displacement is Factor okay so for example let's say you're traveling from point A to point B again point A to point B let's say if I travel North okay and then I turn right travel is to reach my destination from point A to point B this is what we call as distance so let's say here 10 km and then here 15 km this is called distance so total you have traveled a distance of uh 25 km okay that's what we call as distance but if you talk about displacement displacement is the shortest distance so the shortest distance is only one it's only here is the only way the shortest distance so this is what we call as displacement because why Direction matters you cannot go the other way here you cannot go some other direction and come here no it has to be the fastest the shortest way the shortest distance so that is called displacement so in other words for displacement Direction methods so that's the difference between scalar quantity and also vector quantity so let's come back to mathematics okay so in scalar and Vector okay we'll forget about scalar for now we are going to learn about Vector right so in Vector there are a few things you have to know so for example let's say you have again you have point a point B okay this a this is B how do you represent a vector so let's say this you're moving from point A to point B so this Vector you can say let's say they give you Vector a so when you write a vector okay Vector a b you must always put the arrow here because it's telling you that I'm moving from point A to point B okay that's the vector a b if you write b a it's a different direction it means you're moving backward okay so Direction matters okay so make sure you use a b so what's the vector of a b Vector of a B is a and then the curly bracket at the bottom okay make sure you write this if you just write a this is wrong you have to write the curly bracket make sure you write that so this is what we call as Vector but what if I switch Direction now what if I'm traveling from B to a let's say here same thing a b but now I'm traveling from B to it the arrow is going towards a so now your direction let's say it's the same as this okay this is the same as a b so in this case your direction is going to be b a correct so it's going to be opposite direction of this so it's going to be negative a negative a because you're traveling opposite direction okay you can either write here ba or you can also write negative a b same same meaning okay it means you're traveling opposite direction so this is what we call as negative factor okay this is a normal vector and then here's a positive a negative factor now what else you have to learn for the basics so you have to also learn about this the first part the second part you have to learn is what we call as equal Factor what is equal vector equal vector means two vectors have the same direction and the same magnitude so it can be for example let's say I got again two vectors here w x y z okay so now both of these have same magnitude and they have same direction since they have same magnitude and the same direction you can say that Vector WX is equal vector to why Z okay equal vector same magnitude and same direction okay so that is the second thing you have to know the third thing you have to know is parallel vector parallel Vector what is parallel how is that different from equal vector see parallel Vector is they have the same direction because they are moving you know same same direction right but they don't have the same magnitude so for example let's say I got here w x okay moving same direction here same thing I'm moving the same direction but my magnitude is different magnitude in other words is like length Okay it's the length of the vector so sorry this is not w x this should be another different Vector so let's put this Y is that okay so here is a here is B two different vectors okay but they have the same direction so in this case if you write if sometimes the question will ask you to prove that it's parallel Vector when you want to prove make sure you write like this you write the vector as a w x in this case is WX equals to y z correct but I cannot just write equal why because the magnitude is not the same so you will have a constant here K okay you can put K you can put any other alphabet but I'll just use K here so K means a constant that means the difference between parallel the two parallel vectors w x and y z they are the same direction but the the difference is they have different magnitude that's why you have a constant here okay if you don't understand don't worry later we have more examples to work on so you can also write like this a equals to KB is the vector okay I'm using the vector all right so this is parallel Vector now sometimes they can also assume co-linear okay I'll just add here number four collinear what is collinear choline is actually the same as Factor the only difference is sorry colony is the same as parallel Vector the only difference is parallel there are two separate lines but co-linear they're the same line so for example let's say you have a long line here this is X Y Z I can see that X Y is parallel to why is that correct we can say they are parallel but they are you can also say they are collinear y because they have a common point a common point is why right they're sharing the same point so in other words you are going to use the same thing which is this for co-linear but the only difference is you have to prove that they are parallel okay right here proof you have to prove that number one they are parallel and you have to prove that they have common Point that's the difference if I say they are parallel they just have you just have to prove this okay parallel you just have to prove this but if they are collinear you have to prove that they are parallel and you have to also prove that they have common point in this case the Y is the common Point okay so no worries if you don't understand what I'm saying we'll go straight to the examples oh wait there is one more important one which I forgot to add on I need forgot to explain the fifth one that you have to learn is magnitude okay magnitude so I usually they represent magnitude of the vector as modulus you have a modulus here okay this will be the vector and then you have a modulus side by side okay this is what you call as magnitude so for example let's say you are traveling from okay point a okay one person travel North here Point okay 0.01 travel to a the other travel to B okay let's say this is M1 okay the the vector and this is M2 okay Vector one vector two what is their result what is their magnitude so the magnitude between these two uh A and B is here correct so that's why like I said in other words magnitude is basically the length Okay the shortest distance length the magnitude so how do you find the magnitude here so m the magnitude of this Vector is equivalent to M1 square plus M2 square and U square root so what is this this is Pythagoras theorem is that how you spell ISO not too sure anyway I'm not teaching English so that's fine so anyway as long as you understand that magnitude means you're searching you are looking for the link okay so you're basically using pythagore's theorem okay M1 square plus M2 square and a square root okay so these are the basics that you have to know and we'll use all of this in some of the questions later okay so let us get started with the questions the diagram on the right shows Vector F that represents the force applied on an object from J to K find the magnitude and the direction so they ask you to find magnitude and Direction so the direction is from J to K correct so in this case let's start with magnitude so like I said magnitude is basically looking for the the length correct Pythagoras Theorem so in this case how do you find pythagore's theorem so you have here this line here each box is one cm so here is 2cm and then here is what one two three four so four CM so in other words you're going to do 2 square plus 4 squared and then you square root so your answer should be 4.47 Newton please write Newton because this is force right where is it yeah force force is the unit is Newton okay so 4.47 Newton next you want to find for Direction so in this kind of question right usually they want you to solve when they ask for Direction they want you to have the bearing okay because here you see here you have North correct so it's telling you the direction so when you want to prove the direction you have to show how far away from the north okay so in other words since you're traveling from J to K right so if you put a compass here he's North so how far away is the direction I mean the the direction from J to K so in other words you're looking for the angle here okay how far away from North you're traveling okay so with that we'll have to do trigonometric ratio so in this case let me just write it properly for you all so you're going to have to find this angle right like I said so here is north so you want to find this angle so how do you find this angle you're going to use tangent the length is two here is four right so you're going to use tangent let's say this is a Theta equals to opposite is 2 over 4 opposite over adjacent so tangent sorry theta equals to tangent inverse 1 over 2 your answer should be 26.57 okay but please don't leave your answer like this because you're writing bearing so bearing you must always have a three digit number in front so in this case it's going to be 0 to 6.57 degree okay this is how you know me right bearing so that means it is 26.57 degree away from North okay all right next two cars A and B are moving away from town o a moves to the north while car B moves to the east okay find the distance between the two cars after both cars traveled for one hour given that so they gave you the they give you the distance right the magnitude means the length length means the distance they have traveled so in this case the second all right so in this case you have Point O Okay is from point O they are traveling car a moves to the north okay let's say move north here his own wolf North car here I'll give you that right here and then copy moves to the east this is B okay what is the distance so now the the distance they gave you for OA is 90. so here is 90. and then OB is 75. they ask you what is the distance traveled between the two cars so in other words they are asking for what magnitude again right the length the distance between the two that means the length so it's Pythagoras Theorem again so the distance in this case is equals to 90 square plus 75 Square and then you square root so your answer should be one one seven point one five km right so this is a distance between the two cars right next okay the reason why I'm giving you all this is I know some for some of you all might seem very easy but the reason I'm giving is because so that you all are familiar with Vector because Vector right this chapter it might seem easy right because I mean depends on certain people some might find it hard but if you just learn the theory it's actually very easy but when you start solving questions they can mix up the question and make it very complicated so that's the problem with Vector but no worries we'll go through the basic first let me go to go through more complicated questions right so now find pairs of the same Vector so now they ask you for same Vector so what is same Vector same Vector means they must first have same magnitude or same length Okay same meaning or sorry n they must also have same direction that's what makes this the vectors the same must have same magnitude and same direction so in this case let's start with a okay a vector a is the same as which one let's see should be the same as d right it should be the same as D same direction same magnitude both cows three boxes okay next let's see here c c is the same as F correct so C same as f what else b b is the same as um e right so B is the same as e what else um if we are done D we're done yeah right let's see the other one on top so here MN should be the same as CD right same magnitude same direction so m n make sure your arrow up there is the same as CD and then e f should be the same as um let's see KL right no F anymore yes I still have a b a b should be the same as g h okay yeah so that should be so these are all the paths okay same magnitude same direction next the bottom one here State the following vectors in terms of a so they want you to form all the vectors here using only a okay you cannot use any other alphabets okay so use the vector a so in this case you know that let's see what is Vector a of us Vector a is what four boxes moving diagonally right so here one two three four so you know you got four boxes moving up okay moving up four boxes okay now why I say four boxes because we need to use you'll see why I mean why the four is important because you want to form your magnitude okay so in this case four moving upwards so now let's see p p q sorry p q p q is what only moving same direction as a so I can say that it is the same Vector same direction right so same Vector but this one is only two boxes correct it's only two boxes it's half off two over four right so it's half of the original so it's gonna be one over two a okay one over two eight half of a next let's see this vector vector X Vector X is moving opposite direction so you know it's going to be a negative Vector so negative a Okay negative Direction but let's see how many uh boxes the magnet the how many times longer okay so you got one two three four boxes so here and here it should be the same four boxes right and then you got another additional what two boxes here right so that means you got to make it easy right total is what six boxes so it's six over four okay six boxes out of the original four boxes here okay so you simplify this you get three over two okay I don't make it complicated so you just see how many boxes and then you can write in terms of fraction now let's see why Vector y now Vector Y is also moving opposite direction right it's going downward so it's going to be negative a but what's the constant so it's going to be one two three four five six seven boxes okay so it's seven out of four boxes okay um next you're going you're doing RS so RS is same direction is going upward so it's going to be Vector a but the difference is what one two three four five you're going five boxes in front so it's five out of the four boxes original okay so five over four a okay so this is how you form the different Vector you compare with the original ones okay I'm just giving you you guys some basic practice so that you all understand the bigger picture later right so let's go to the next question given that a b equals to 5 a p q equals to twenty a Express a b in terms of p q if a b is parallel to P Q you see now we're going to solve parallel questions so they say a b is parallel to P Q so remember I told you if it's parallel you must prove that they have constant right they have a constant between them this K so you have to prove that they have a constant so in this case you know that a b is equals to 5A correct and you have p q p q equals to 20 a so this is basically what it's like simultaneous right you're trying to fill up what is in the a so in this case I'm going to shift the a I want to find a what is factor a p q over 20 equals to A okay now since I already know what is a I can substitute into the other equation right it's like simultaneous this equation one is equation two then you're gonna do substitution method so a b is equals to 5 times a is what p q over 20. so a b is what a b is equals to so here you can simplify you get 4 so 1 over 4 of P cubed so see you can prove the format right equals to k p q so what is k K is actually 1 over 4 right so since you can prove this right you can prove this so therefore the apparel right so therefore uh this is the question is asking what Express a b in terms of p q so this answer okay here's your answer no need to show all this all right this I'm just trying to explain to you okay once you can prove that this thing exists that means they are parallel okay all co-linear depending on the question okay so this is the parallel question next show that point l m n are collinear given that LM equals 6X MN equals to 18x so now they want you to prove that it is collinear so co-linear is the same you have to prove that they are parallel and then you have to prove they have common point okay common point so now since they ask you to prove LMN that they give you l m and MN so if I want to show that they are parallel I can use MN should be equals to k m n right you can we have to prove this first okay so let's prove that so LM you know is 6 x okay similar to what how we did the previous question and then MN is equals to 18x so I'm going to shift I want to find the X first so I can substitute to the other equation so LM o six equals to X okay so this is your first equation this is your second equation so you're going to substitute first equation into second so you get m n equals to 18 times the x value is LM over 6 okay so MN is equals to you can simplify you get 3 l m okay once you can prove this you know that it is parallel here is parallel next they also have common Point what are the common points there they both have M so in other words it's something like this you have l you have let's say here is M and here is n okay so they have common Point common point is M so you can write since MN equals to 3 LM and both have common point m therefore the three points points l m and n are co-linear all right so the same thing you just have to prove that they have they are parallel and you want to prove they have common Point that's it so the common point is here so this is how you solve co-linear questions all right next given that non-zero Vector u and v are not parallel find the value of M and N for each of the following so in this case when I say non-linear no sorry non-zero Vector they are just saying that it has value that means it has magnitude it's not a zero Factor okay so it's a it has magnitude so this kind of question right what you're going to do since they're not parallel you can shift all to one side okay shift all to one side this is how you're going to solve this kind of question okay M for M plus three U H shift this to the other side so you get minus n minus 7 V equals to zero so you want to find the value of M and N correct so you just take this equals 0 and this equals to zero as well so 4M plus 3 equals 0 so m is equals to negative three over four and then the other one is negative n minus 7 V sorry there's no need to write V let's write this it equals to zero so n is equals to what negative 0 is still 0 and then here plus seven so this is your answers all right so let's see question B same thing you move all to one side so there is shift for you so you just have to write what M plus n minus one actually don't need records equals to zero so this would be equal to zero and this one also equals to zero so here you have two unknown so it's not a straightforward as question a because question a is only one unknown right so you go two unknown so you have to use simultaneous so in this case negative this first equation is this second equation is negative M minus 2 N minus ten equals to zero so M minus two n minus ten equals to zero so what I'm going to do is I'm going to shift this m equals to 1 minus n okay I change this make this my first equation this is my second equation oh wait can I do elimination you know what I'm going to do elimination it's going to be so much easier okay so in this case sorry M minus 2N equals to ten this one is M plus n equals to 1. okay so I'm going to do elimination so I take equation one equation two so when um 2 minus 1 what do I get negative 2 minus so negative 3 n equals to 9 N equals to negative 3. so what is my what is my M value so M plus n equals to 1 right so m equals to 1 minus n n is negative three answer is four given that X Y and Y W are parallel vectors now they give you what modulus X Y so this is magnitude magnitude of X Y is six magnitude of VW is 21 Express VW in terms of X Y so again they want you to prove that they are parallel correct so you can either prove it's up to you how you want to prove it as long as you prove that they have a constant all right but in this case I'm not going to use this I'm going to use the opposite I'm going to change it like this I'm going to use VW equals to k x y why am I doing this because the question is asking to express VW not express X Y so I'll make VW as my subject so in this case how do you solve same thing you're going to use that write this now if they are give you they give you what modulus right so you just add modulus so if you add modulus here here so you have to add modulus now for a modulus the constant right it doesn't have to stay inside the constant can come out so when a constant comes out it doesn't really affect so the K comes out and you have X Y here okay now you can substitute what is VW VW is 21. K you don't know X Y is six so what is k K is 21 over 6 correct 21 over 6 so you can simplify this so K is equals to 7 over 2 right you can divide by three so 702 so once you have found your K value you can just rewrite your they asked to express VW right so VW is equals to 7 over 2 x y so this is your answer all right next now we have already covered the basic so now let us look at addition and subtraction of vector so for addition and subtraction of vector there are two types okay the first type is when you see they are only parallel vectors okay if parallel vectors they are much easier why because for example if you do addition right for parallel vectors if you have a vector here let's say this is a okay and then you plus another parallel Vector let's say this one is um two two a okay let's say this 3A this 2A so the resultant Vector you get is the combination of the two lines so you get 5A okay is very straightforward but if it is a subtraction subtraction is let's say you got here 3A minus 2A so in the end your resultant will be a okay so this is subtraction however you this is the simple one but sometimes they'll ask you non parallel Vector okay for non-parallel Vector there are three laws that you have to know the first law is called triangle law okay triangle law is such that for example let's say you have here one point here let's say a B and C here okay so you're traveling from point A to point B to point C so point A to point B is let's say a here point B to C let's say the vector is B okay so what is your resultant Vector so resultant Vector means like from a to c so a to c is basically a plus b correct because if you want to go from a to c you have to go through uh a b and then BC correct so that's your A and B so your vector is a b however sometimes the question can be tricky how they will ask you like this let's say they give you a same thing here B and then here is C however here the vector is a but the vector B is opposite direction so how do you find your resultant vector your resultant Vector if let's say you are going from Ace a to c right so a to c is basically what you have you have to go through a b and then BC correct same as the first part A the this one same as this you go b a b and then BC so what is a b what's the vector of a b a right what about BC now BC this Vector is C to B now you want B to C so you have to use negative vector so your answer is a minus B okay so this is your resultant Vector from a to from a to c okay use a minus B okay so this is how you the main thing is you have to follow have that Chain Reaction you know that chain that flow that means if you want to go from a to c make sure you have a to B and then B to C like that you cannot just jump a to c okay never mind we'll do more questions and you understand what I'm saying okay next the second law is parallelogram law okay parallelogram law is actually quite similar to Triangle as well but the difference is here if let's say here is a B then here is uh C d let's say you're given this one is a here is B if you want to travel from a to c sorry a to c the resultant Vector here is what a plus b how do you know reason is because since here is a and here is parallel right these two are parallel right so that means the vector here is also a because they have the same length and same direction the same magnitude and same direction so in other words it's also a so when you form your Vector AC you're going through a d and DC correct so is B plus a or you can write a plus b same thing okay so this is what we call as parallelogram law the most common one which you will use is this actually not say most common all three you use is very common actually so this is called polygon polygon law okay so polygon law obviously there is a polygon so oops sorry so you've got a polygon here let's say I have a pentagon a b c d e let's say A to B here the vector is a B C d and e is opposite okay now if you want to find vector AE okay Vector AE is what you have to go through a to B and then B to C and then C to D and then D to E correct you notice they all have something in common here they have in common here they have in common here you have in common and then the ending here and here is this AE all right so I hope you take note of that so that when you form your when you form your equation you get you use the correct chain right so now what is a e a e is the vector is E equals to a plus b plus C plus d right so this is polygon law so you are actually following the the chain right that the chain that I said is now as long as you follow the correct pathway okay you're following the flow you'll get the correct answer right because as it gets more complicated your the like you say like you see here I form this equation correct when when the question becomes more complicated your equation may not be the same as your friend's equation okay he might have some other equation your other friend might have some other equation but if all of y'all are following the uh the correct route okay the correct route maybe he followed different route you follow different route in the end you will still get the same answer okay now if you don't understand don't worry we'll go straight to examples Vector P represents the velocity of 17 kilometer per hour to the South and Vector Q represents velocity of 80 kilometer per hour to the east find the direction and magnitude of the resultant Vector so they gave you the results and Vector is p plus Q correct so P plus Q where is p p is 70 kilometer power to the South so let's assume we start here to the South okay let's go down 70. a kilometer per hour next plus Q where is q q is the velocity of 80 kilometer per hour to the east so that means to the east this is 80 kilometer per hour all right find the direction and magnitude of the resultant Vector so same thing you want to find magnitude means here right this one so magnitude of p plus Q so magnitude here in this case is going to be right here let me make magnitude or maybe I should write P plus Q okay because it's p right this is q so P plus Q with the magnitude you should get 70 squared plus a t squared and then you square the whole thing your answer should be one zero six point three zero kilometer per hour okay next they want Direction so Direction like I said whenever they ask this kind of question it is usually the bearing right direction means the bearing so you want to find if we put a Compass here this is North up here is north what is the the angle so you want to find this angle here from North all the way to the direction you're moving okay P plus Q so this is p plus Q okay so you want to find that angle so to find that you must first find out this angle right so we can use the trigonometric ratio again so tangent let's assume this is uh data okay let's this is Theta so you want to use tangent Theta sorry theta equals to opposite is 80 over adjacent is 70. so what is Theta Theta is tangent inverse 80 divided by 70. answer should be 48.81 sorry so if you want to find your direction so you have to take 180 because you want to find this direction right not the Theta so direction should be 180 minus 48.81 so your answer is one three one point one nine so this is your Direction so that means you're moving 131.19 degree away from North okay so that's how you solve this question next given that ABCD is a trapezium with 3 a b equals to 2B 2 DC Express the following in terms of X and Y so question number one you want to find a b so a b is here correct now there is no way for you to find because first of all you know you can find you can use a d plus DC but you don't know what is CD so you cannot use what is given here but they give you this so you can use this so for in this case you can say that 3 a b equals to 2 DC so what is a b a b is equals to 2 over 3 what is DC this is given to you which is y so write back y so this is answer for a question B question B they ask for AC so AC so Vector AC is equals to act Vector a d plus DC because why because it's given to you so I'm going to just use whatever they gave me a d is X sorry it's negative X because it's opposite direction Negative X Plus y now question C you're looking for BC BC is here from here to here so there are few ways you can solve this array the first way is you use ba plus a d plus DC you can do that okay because all of that is given a b is you already found the answer from question a right so you can just use that to solve another way you can use is use a b Plus AC AC you've already found here question B right so you can actually use both both methods so we'll give you the same answer so let us do that together so I'm going to use the first method okay so you're looking for BC right so BC is equals to b a plus a d plus DC okay so I'm going to take this long Round Here So ba is what ba is the opposite of a b right opposite so it's negative Vector negative 2 over 3y plus a d is negative X because X is d a right you're looking for a d so it's opposite X opposite of the actual Vector so it's negative X and then plus DC is y right so you should get 1 over 3y minus X okay so that's BC now you want to find third one and sorry fourth one b d where is BD BD is here to here okay so it's up to you you can use here BC Plus CD or you can use a b sorry B A Plus uh a d right both methods is acceptable so now BD I'm going to use b a plus a d So ba is negative 2 over 3y and then a d is negative X so yeah that's it that's the answer for BD right next given o x y and z are four points with okay so to give you o x o y and o z Vector if points x y z are collinear say collinear find the value of K so in this case collinear means what it has to be parallel and it has to be it has to have the same the common point one common point correct so how do you find parallel so now you got X Y and Z right you want to prove that these three are collinear so you can use either you can it's up to you right you can use any Vector that you want as long as it's between these three points so I'm going to use x y equals to um exit okay exit is up to you you can use x y equals to y z you can use a y z equals to x z you know is entirely up to you in this case I'm just going to use x y equals to XZ so here I'm going to put a constant but wait I'm not going to use K reason is because K is already used here so I don't want to get confused so I use another alphabet so I'm going to use say a okay I'm going to use a as a constant so in this case you need to find what is X Y first x y equals to you don't have X Y you have X you have ox and o y so what you can do is you can do x o plus o y okay XO plus o y like I said if these two is uh in common right if they have the same one that means you can combine so in the end you still get X Y okay so XO is opposite of ox so it's negative 4 x minus 2y Plus or Y is what k x minus y so you will have negative 4 plus k plus 2 minus 1 y so here is negative 4 plus k X Plus y okay so you've already found your X Y okay now you want to find x z the second part here okay so what is XZ XZ is same thing you're going to use you're only given o x and o z correct so you can straight away use the uh ox and Oz Vector so it's going to be x o plus o z okay so what is XOXO is same thing negative 4X minus 2y and then uh plus o z is what 6x plus 5y so you should get negative four plus six X plus 2 plus 5 y so it's negative sorry it's 2X plus 7y okay so now you can straight away substitute into this equation here so you get X Y is what let me change color right X Y is negative 4 plus k X Plus y equals to A and then XZ is some 2x plus 7 y okay so I'm going to expand this so I will get negative 4 plus k X Plus y equals to 2A X plus seven a y okay so from here you can compare the x value with the x value and the Y value with the Y value so in this case negative 4 plus k equals to 2A is your first equation second equation is here is coefficient is one right so you're going to use 1 equals to seven a so what's the value of a 1 over 7 correct this is the second equation so one over seven so you can substitute into your a value here to find the K value so what is K value K value is equals to 2A a a is one over seven plus four so two over seven plus four so your answer is 30 over 7. all right so that's how you solve this kind of question all right let's see next okay this is the last question here the diagram on the right shows the plan of Alice of a residential area which forms a quadrilateral ABCD there is a Lamppost at position e okay so there's a Lamppost here position e and oh where b e to e d is three to one so b e is here so it's three to one okay three two one l e a b and d c are parallel okay A D and DC is parallel and then they give you this DC equals to four over three a b so now first question what they want Press B D and AE in terms of X and Y so where is b d b d is um here right the diagonal line here so you want to do that you can still reform your vector first v d sorry [Music] BD so how can you form the vector BD is the same as taking here BA Plus a d correct so I'm going to do that b a plus a d so B A is negative 24 x plus 80 is 20 y so this is your answer right next you want to find a e so where is a e a e is here so there are few ways you can solve it's either you move very easier right so it's either you take a d plus d e or you take a B plus b e okay both ways work okay like I said as long as you follow the chain the the flow of the the correct route to get to your destination you will get the same answer let's say like I said some people will get if your friends might get a different equation compared to you but in the end you still get the same answer as long as you obey the flow okay so now let's solve this together AE I'm going to use a B plus b e okay so a b is given to you 24 x but what is b e b e is you see they gave you the ratio three to one right three to one so this one is three out of the whole thing is what out of four correct because b e is 3 and e d is one right so that means total is four so that means b e is three out of four of the total length Okay so three out of four means what three over 4 of BD the total length is BD right so what is BD you have already found BD here so you just use that 24x plus 3 over 4. times negative 24x plus 20y okay so 24x minus so 4 18x here you get 5 15. why okay so 24 okay wait or 24 minus 18 you should get six X Plus 15y okay so this is AE yeah that's it all right let's look at question B so question B they want you to show that the Le a e is parallel to lebc so to prove that they are parallel you have to write the I told you the condition right AE equals to k b c if you can prove this that means there parallel so in this case you have already found AE AE is here correct you've already found AE next you want to find BC so BC is from here to here so the only way is actually there are many ways you can use here and then here or you can use the long way also can it's up to you so I'm gonna use um BD plus DC okay so where can you find DC DC is given here so just use this okay so okay b d is what again negative 24 X Plus 20y plus DC is what four over three a b where's a b well a B is 24x okay so negative 24x plus 20y plus um 32. X okay so you should get 8X plus 20 y okay so now you have already found AE and your BC okay I'm gonna write this side okay because I got no space there yeah let me just right here so in this case a e equals to k b c correct so a is what six X Plus 15y equals to K b c is um 8X plus 20y so to prove that they have a constant you must first make them become the same Vector so they can simplify so I'm going to factorize um 3 here so I should get 2x plus 5y okay and then on the opposite side I want to factorize 4. okay remove four I should get 2X Plus 5y right so if you notice see both side same Vector correct so because they are the same you can straight away cancel them because why if you bring to the opposite side it becomes divide right so you still can cancel so in the end you get what you get 3K 3 equals to 4K so what is k k equals to 3 over 4 so that means a e is equals to three over four BC so once you have proven this that means they are parallel so that is all for this video These are some of the main subtopics that we have covered the basics of vector as well as addition and subtraction of vector all right so I hope you find this useful and if you did find it useful don't forget to like share subscribe and also share it with someone who will benefit from this right so that is all for today I hope you find it useful and I will see you on my next video take care bye